Off-contract at the end of the year and eligible for unrestricted free agency, Giants captain Greene has yet to make his call on his future.
Geelong have emerged as a possible suitor for the 32-year-old Victorian product, who has played all 274 games in his career at GWS.
Greene has also previously mused on a future as a one-club player, having surpassed Callan Ward as the club's all-time games record holder when he played his 268th game.Â
The Giants have previously secured signatures of star players before the final season of their respective contracts, having done so with Sam Taylor, Connor Idun, Toby Bedford, Finn Callaghan and Aaron Cadman.
But Kingsley isn't putting any pressure on Greene.
"I'd prefer him to sign today, but the reality is that's not going to happen," Kingsley said.
"He needs time to discuss and think, so we'll give him as much time as he needs.
"Again, I'm not going to press him for information. I'm not going to be in his ear every day.
"He's a big boy. He can make big boy decisions, and that's what he'll do when he feels comfortable doing that.
"Whenever that is, we're comfortable with what his decision will be."
Greene is just one of many key players who are off-contract at the end of the season.
Star key forward Jesse Hogan is the biggest name, alongside inaugural Giants player Stephen Coniglio.
Among them also are fellow tall forwards Jake Riccardi and Max Gruzeweski, defender Conor Stone, and rucks Kieren Briggs and Nick Madden
"I'm really comfortable with where we're at," Kingsley said.
"We're in a really strong position not only in the immediate future but beyond that, so I'm confident that those guys will sign.
"But again, there's things that bob up, and that might happen again, who knows? I feel really confident now at least.
"Those guys are influential players for us, so we're certainly very hopeful."
Kingsley backed Hogan to rediscover his goalkicking prowess after struggling against St Kilda's Jack Silvagni last week.
Out of action for a month with a hip injury, 31-year-old Hogan kicked just one goal in the final quarter as the Giants lost by eight points.
"He pulled up well. There's nothing physically that's hindering him at the moment so that's great," Kingsley said.
"His first game back after four or five weeks presented a few challenges, and we were hopeful that he would be able to overcome them a little easier than what he did.
"That's the nature of the game."
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